The category of reduced imaginary Verma modules

Let $A=(a_{ij})_{0\leq i,j\leq N}$ be a generalized affine Cartan matrix over $\mathbb{C}$ with associated affine Lie algebra $\hat{\mathfrak{g}}$ and Cartan subalgebra $\hat{\mathfrak{h}}$. Let $\Pi=\{\alpha_{0},\alpha_{1},\cdots,\alpha_{N}\}$ be the set of simple roots, $\delta$ the indivisible imaginary root and $\Delta$ the root system of $\hat{\mathfrak{g}}$. A subset $S\subseteq\Delta$ is a closed partition if for any $\alpha,\beta\in S$ and $\alpha+\beta\in\Delta$ then $\alpha+\beta\in S$, $\Delta=S\cup(-S)$ and $S\cap(-S)=\emptyset$. The classification of closed partitions for root system of affine Lie algebras was obtained by H. Jakobsen and V. Kac in [9] and [10] and independently by V. Futorny in [3] and [5]. They show that closed partitions are parameterized by subsets $X\subseteq\Pi$ and that (contrary to what happens in the finite case) there exists a finite number (greater than 1) of inequivalent Weyl group orbits of closed partitions. When $X=\Pi$ we get that $S=\Delta_{+}$ and we can developed the standard theory of Verma modules, but in the case $X\subsetneq\Pi$ we obtain new Verma-type modules called non-standard Verma modules.

The theory of non-standard Verma modules was initiated by V. Futorny in [4] (see also [6]) in the case $X=\varnothing$ and continued by B. Cox in [1] for arbitrary $X\subsetneq\Pi$. The case $X=\varnothing$ give rise to the natural Borel subalgebra associated to the natural partition $\Delta_{\textrm{nat}}=\{\alpha+n\delta\ |\ \alpha\in\Delta_{0,+}\ ,\ n\in\mathbb{Z}\}\cup\{k\delta\ |\ k\in\mathbb{Z}_{>0}\}$. The Verma module $M(\lambda)$, of highest weight $\lambda$, induced by the natural Borel subalgebra is called imaginary Verma module for $\hat{\mathfrak{g}}$, when it is not irreducible it has an irreducible quotient called reduced imaginary Verma module. Unlike the standard Verma modules, imaginary Verma modules contain both finite and infinite dimensional weight spaces. Similar results hold for more general non-standard Verma modules.

In [2], while studying crystal bases for reduced imaginary Verma modules of $\hat{\mathfrak{sl}_{2}}$, it was consider a suitable category of modules, denoted $\mathcal{O}_{red,im}$, with the properties that any module in this category is a reduced imaginary Verma module or it is a direct sum of these modules. In this paper, by appropriate modifications we first define a category $\mathcal{O}_{red,im}$ for any affine Lie algebra and we show that all irreducible modules in this category are reduced imaginary Verma modules and, moreover, that any arbitrary module in $\mathcal{O}_{red,im}$ is a direct sum of reduced imaginary Verma modules.

It should be noted that the results presented in this paper hold for both untwisted and twisted affine Lie algebras. The paper is organized as follows. In Sections $2$ and $3$, we define, set the notations and summarize the basic results for affine algebras, closed partitions and imaginary Verma modules. In section $4$ we introduce the category $\mathcal{O}_{red,im}$ and present some of its properties. Finally, in section $5$ we present the main results of this paper.

In this section we fixed some notation and the preliminaries about affine algebras and root datum are set up.

Let $S$ be a closed partition of the root system $\Delta$. Let $\hat{\mathfrak{g}}$ be the untwisted affine Lie algebra which has, with respect to the partition $S$, the triangular decomposition $\hat{\mathfrak{g}}=\hat{\mathfrak{g}}_{S}\oplus\hat{\mathfrak{h}}\oplus\hat{\mathfrak{g}}_{-S}$, where $\hat{\mathfrak{g}}_{S}=\bigoplus_{\alpha\in S}\hat{\mathfrak{g}}_{\alpha}$ and $\hat{\mathfrak{h}}=\mathfrak{h}\oplus\mathbb{C}c\oplus\mathbb{C}d$ is an affine Cartan subalgebra. Let $U(\hat{\mathfrak{g}}_{S})$ and $U(\hat{\mathfrak{g}}_{-S})$ be, respectively, the universal enveloping algebras of $\hat{\mathfrak{g}}_{S}$ and $\hat{\mathfrak{g}}_{-S}$.

Let $\lambda\in P$. A weight $U(\hat{\mathfrak{g}})$-module V is called an $S$-highest weight module with highest weight $\lambda$ if there is some non-zero vector $v\in V$ such that:

• •

  $u\cdot v=0$ for all $u\in\hat{\mathfrak{g}}_{S}$.

• •

  $h\cdot v=\lambda(h)v$ for all $h\in\hat{\mathfrak{h}}$.

• •

  $V=U(\hat{\mathfrak{g}})\cdot v\cong U(\hat{\mathfrak{g}}_{-S})\cdot v$.

In what follows, let us consider $S$ to be the natural closed partition of $\Delta$, i.e., $S=\Delta_{\textrm{nat}}$ and so $\mathfrak{b}_{\textrm{nat}}=\hat{\mathfrak{g}}_{\Delta_{\textrm{nat}}}\oplus\hat{\mathfrak{h}}$. We make $\mathbb{C}$ into a 1-dimensional $U(\mathfrak{b}_{\textrm{nat}})$-module by picking a generating vector $v$ and setting $(x+h)\cdot v=\lambda(h)v$, for all $x\in\hat{\mathfrak{g}}_{\Delta_{\textrm{nat}}}$ and $h\in\hat{\mathfrak{h}}$. The induced module

is called an imaginary Verma module with $\Delta_{\textrm{nat}}$-highest weight $\lambda$. Equivalently, we can define $M(\lambda)$ as follows: Let $I_{\Delta_{\textrm{nat}}}(\lambda)$ the ideal of $U(\hat{\mathfrak{g}})$ generated by $e_{ik}:=e_{i}\otimes t^{k}$, $h_{il}:=h_{i}\otimes t^{l}$ for $i\in I_{0}$, $k\in\mathbb{Z}$, $l\in\mathbb{Z}_{>0}$, and by $h_{i}-\lambda(h_{i})\cdot 1$, $d-\lambda(d)\cdot 1$ and $c-\lambda(c)\cdot 1$. Then $M(\lambda)=U(\hat{\mathfrak{g}})/I_{\Delta_{\textrm{nat}}}(\lambda)$.

The main properties of this modules, which hold for any affine Lie algebra, were proved in [6] (see also [7] for more properties on this modules), we summarize them in the following.

Suppose now that $\lambda(c)=0$ and consider the ideal $J_{\Delta_{\textrm{nat}}}(\lambda)$ generated by $I_{\Delta_{\textrm{nat}}}(\lambda)$ and $h_{il}$, $i\in I_{0}$ and $l\in\mathbb{Z}\setminus\{0\}$. Set

Then $\tilde{M}(\lambda)$ is a homomorphic image of $M(\lambda)$ which we call reduced imaginary Verma module. The following is proved in [6], Theorem 1.

Consider the Heisenberg subalgebra $G$ which by definition is

We will say that a $\hat{\mathfrak{g}}$-module V is $G$-compatible if:

1. (i)

   $V$ has a decomposition $V=T(V)\oplus TF(V)$ where $T(V)$ and $TF(V)$ are non-zero $G$-modules, called, respectively, torsion and torsion free module associated to $V$.

2. (ii)

   $h_{im}$ for $i\in I_{0}$, $m\in\mathbb{Z}\setminus\{0\}$ acts bijectively on $TF(V)$, i.e., they are bijections on $TF(V)$.

3. (iii)

   $TF(V)$ has no non-zero $\hat{\mathfrak{g}}$-submodules.

4. (iv)

   $G\cdot T(V)=0$.

Consider the set

We define the category $\mathcal{O}_{red,im}$ as the category whose objects are $\hat{\mathfrak{g}}$-modules $M$ such that

1. (1)

   $M$ is $\hat{\mathfrak{h}}^{*}_{red}$-diagonalizable, that means,

   $$M=\bigoplus_{\nu\in\hat{\mathfrak{h}}^{*}_{red}}M_{\nu},\mbox{ where }M_{\nu}=\{m\in M|h_{i}m=\nu(h_{i})m,dm=\nu(d)m,i\in I_{0}\}$$

2. (2)

   For any $i\in I_{0}$ and any $n\in\mathbb{Z}$, $e_{in}$ acts locally nilpotently.

3. (3)

   $M$ is $G$-compatible.

4. (4)

   The morphisms between modules are $\hat{\mathfrak{g}}$-homomorphisms

Recall that a loop module for $\hat{\mathfrak{g}}$ is any representation of the form $\hat{M}:=M\otimes\mathbb{C}[t,t^{-1}]$ where $M$ is a $\mathfrak{g}$-module and the action of $\hat{\mathfrak{g}}$ on $\hat{M}$ is given by

for $x\in\mathfrak{g}$, $m\in M$ and $k,l\in\mathbb{Z}$. Here $x\cdot m$ is the action of $x\in\mathfrak{g}$ on $m\in M$.

Proof. Let $M\in\mathcal{O}$ and let $\hat{M}$ be its associated loop module. If $M$ is finite dimensional, it is a direct sum of finite dimensional irreducible $\mathfrak{g}$-modules, and these have highest weights which are non-negative integers when evaluated in $h_{i}$ for any $i\in I_{0}$. So, condition (1) is not satisfied and $\hat{M}$ does not belongs to $\mathcal{O}_{red,im}$. Assume now that $M$ is an infinite dimensional $\mathfrak{g}$-module. Note that condition (2) is satisfied as $\mathfrak{n}$ acts locally nilpotently on $M$. If condition (1) does not hold, we are done. Suppose that (1) holds and that $\hat{M}$ is $G$-compatible. We have $\hat{M}=T(\hat{M})\oplus TF(\hat{M})$ satisfying (i) - (iv) above. Take any nonzero element $\sum_{i=-k}^{k}m_{i}\otimes t^{i}\in T(\hat{M})$ with $m_{i}\in M_{\mu}$ for some weight $\bar{\mu}\in\hat{\mathfrak{h}}^{*}_{red}$. Then by (iv) we have

where $j\in I_{0}$, $r\in\mathbb{Z}\setminus\{0\}$. Hence $\bar{\mu}(h_{j})=0$, for any $j\in I_{0}$, which contradicts to the fact that $\bar{\mu}\in\hat{\mathfrak{h}}^{*}_{red}$. Then $T(\hat{M})=0$ and $\hat{M}=TF(\hat{M})$ which is a $\hat{\mathfrak{g}}$-module contradicting (i) and (iii), and thus (3). This completes the proof.

In this section we will show that the category $\mathcal{O}_{red,im}$ is a semisimple category having reduced imaginary Verma modules as its simple objects. First we will show that reduced imaginary Verma modules have no nontrivial extensions in $\mathcal{O}_{red,im}$.

Proof. Let $M$ be an extension of $\tilde{M}(\lambda)$ and $\tilde{M}(\mu)$ that fits in the following short exact sequence

Suppose $\mu=\lambda+k\delta-\sum_{i=1}^{N}s_{i}\alpha_{i}$, for $s_{i}\in\mathbb{Z}$ and $k\in\mathbb{Z}$, and all $s_{i}$’s have the same sign or equal to $0$. First, consider the case when $s_{i}=0$ for all $i\in I_{0}$. Then $\mu=\lambda+k\delta$ and so, in $M$ there will be two vectors $v_{\lambda}$ and $v_{\mu}$ of weights $\lambda$ and $\mu$ respectively, annihilated by $\mathfrak{n}\otimes\mathbb{C}[t,t^{-1}]$. Moreover, because of the condition (iv) in the definition of $G$-compatibility, these two points are isolated. So, $v_{\lambda}$ and $v_{\mu}$ are highest weight vectors, each of which generates an irreducible subrepresentation (isomorphic to $\tilde{M}(\lambda)$ and $\tilde{M}(\mu)$ respectively), and the extension splits. Hence, we can assume that not all $s_{i}$ are equal to zero and that the map $\iota:\tilde{M}(\lambda)\to M$ in the short exact sequence is an inclusion. Assume that $s_{i}\in\mathbb{Z}_{\geq 0}$ for all $i$.

Let $\overline{v}_{\mu}\in M$ be a preimage under the map $\pi$ of a highest weight vector $v_{\mu}\in\tilde{M}(\mu)$ of weight $\mu$. We have $(\mathfrak{n}\otimes\mathbb{C}[t,t^{-1}])v_{\mu}=Gv_{\mu}=0$, and we are going to show that $G\overline{v}_{\mu}=0$. Assume that $\overline{v}_{\mu}\notin T(M)$. Then we claim that $T(M)=\mathbb{C}v_{\lambda}$. Indeed, we have $\mathbb{C}v_{\lambda}\subset T(M)$. If $u\in T(M)\setminus\mathbb{C}v_{\lambda}$ is some nonzero weight element, then $G\cdot u=0$ and $\pi(u)$ belongs to $T(\tilde{M}(\mu))=\mathbb{C}v_{\mu}$. If $\pi(u)=0$ then $u\in\tilde{M}(\lambda)$ which is a contradiction. If $\pi(u)$ is a nonzero multiple of $v_{\mu}$, then $u$ has weight $\mu$ and thus $u$ is a multiple of $\overline{v}_{\mu}$ which is again a contradiction. So, we assume $T(M)=\mathbb{C}v_{\lambda}$.

Note that for any $i\in I_{0}$ and $m\in\mathbb{Z}\setminus\{0\}$ we have

Then $h_{im}\overline{v}_{\mu}\in\tilde{M}(\lambda)$. Suppose there exists $j\in I_{0}$ such that $h_{jm}\overline{v}_{\mu}\neq 0$ for $m\in\mathbb{Z}\setminus\{0\}$. Because $h_{jm}\overline{v}_{\mu}\in\tilde{M}(\lambda)$ and has weight $\mu+m\delta$, it belongs to $TF(\tilde{M}(\lambda))$. Hence, there exists a nonzero $v^{\prime}\in\tilde{M}(\lambda)$ of weight $\mu$ such that $h_{jm}\overline{v}_{\mu}=h_{jm}v^{\prime}$. Hence, $h_{jm}(\overline{v}_{\mu}-v^{\prime})=0$ implying $\overline{v}_{\mu}-v^{\prime}\in T(M)\cong\mathbb{C}v_{\lambda}$. Then $\overline{v}_{\mu}-v^{\prime}=p\ v_{\lambda}$, for some $p\in\mathbb{C}$. Comparing the weight we arrive to a contradiction. Hence, $h_{in}\overline{v}_{\mu}=0$. So, we get $G\overline{v}_{\mu}=0$.

Recall that the operators $e_{im}$ acts locally nilpotently on $\tilde{M}(\lambda)$. We claim that $e_{im}\overline{v}_{\mu}=0$ for all possible $i$ and $n$. Indeed, assume that $e_{jm}\overline{v}_{\mu}\neq 0$ for some $j\in I_{0}$ and some integer $m$. Then $e_{im}\overline{v}_{\mu}\in\tilde{M}(\lambda)$. Consider the $\hat{\mathfrak{sl}}_{2}$-subalgebra $\mathfrak{s}(j)$ generated by $f_{jn},e_{jn}$ and $h_{jl}$ for $n,l\in\mathbb{Z}$. Let $M_{j}$ be an $\mathfrak{s}(j)$-submodule of $M$ generated by $\overline{v}_{\mu}$. Then $M_{j}$ is an extension of reduced imaginary Verma $\mathfrak{s}(j)$-modules, one of which of highest weight $\mu$. Since $M\in\mathcal{O}_{red,im}$, we immediately see that $M_{j}$ is an object of the corresponding reduced category $\mathcal{O}_{red,im}(\mathfrak{s}(j))$ for $\mathfrak{s}(j)$. But this category is semisimple by [2]. Hence, $e_{im}\overline{v}_{\mu}=0$ for all $i$ and $m$. Therefore, $\overline{v}_{\mu}$ generates a $\mathfrak{g}$-submodule of $M$ isomorphic to $\tilde{M}(\mu)$ and the short exact sequence splits.

Assume now that $s_{i}\in\mathbb{Z}_{\leq 0}$ for all $i$ and not all of them are $0$. As $\tilde{M}(\mu)$ is irreducible and $\tilde{M}(\lambda)$ is a $\mathfrak{g}$-submodule of $M$, the short exact sequence splits completing the proof.

Proof. Let $M$ be an irreducible module in $\mathcal{O}_{red,im}$. As a $G$-module, $M\cong T(M)\oplus TF(M)$ where both summands are non-zero. Let $v\in T(M)$ be a non-zero element of weigh $\lambda\in\hat{\mathfrak{h}}_{red}^{*}$. Then $h_{im}v=0$ for all $i\in I_{0}$ and all $m\in\mathbb{Z}\setminus\{0\}$. For each $i\in I_{0}$ let $p_{i}\in\mathbb{Z}_{>0}$ be the minimum possible integer such that $e_{i0}^{p_{i}}v=0$. If all $p_{i}=1$ we have $e_{i0}v=0$ and then, because $[h_{in},e_{i0}]=2e_{in}$ we get that $e_{in}v=0$ for all $i\in I_{0}$ and $n\in\mathbb{Z}\setminus\{0\}$. Hence, we have an epimorphism $\tilde{M}(\lambda)\twoheadrightarrow M$, since $\lambda\in\hat{\mathfrak{h}}_{red}^{*}$, $\tilde{M}(\lambda)$ is simple and so $M\cong\tilde{M}(\lambda)$.

On the other hand, assume there exists at least one $p_{i}$ such that $p_{i}>1$. We are going to construct a set of elements in $M$ which are killed by $e_{i0}$ for all $i\in I_{0}$. First of all, set $p^{(1)}=\max\{p_{i}|i\in I_{0}\}$ and set $w_{i}:=e_{i0}^{p^{(1)}-1}v$. Note that $w_{i}=0$ if $p^{(1)}>p_{i}$ and $w_{i}\neq 0$ if $p^{(1)}=p_{i}$, so at least one $w_{i}$ in non-zero. If for all $j\in I_{0}$, $e_{j0}w_{i}=0$ we are done, if not there exists numbers $p_{ij}\in\mathbb{Z}_{>0}$ such that $e_{j0}^{p_{ij}}w_{i}=0$ and some of the $p_{ij}$ are strictly bigger than $1$. Set $p^{(2)}=\max\{p_{ij}|i,j\in I_{0}\}$ and set $w_{ij}=e_{j0}^{p^{(2)}-1}w_{i}$, note that at least one $w_{ij}$ is non-zero. If $e_{k0}w_{ij}=0$ for all $k\in I_{0}$ we are done, if not we repeat the process. Because of the locally nilpotency of the $e_{l0}$ for $l\in I_{0}$, in finitely many steps, let say $\ell$ steps, we can find at least one non-zero element $w_{\bf i}$, for ${\bf i}=i_{1}i_{2}\ldots i_{\ell}$ a string of elements in $I_{0}$ such that $e_{l0}w_{\bf i}=0$. Moreover, if ${\bf i}^{-}$ denotes the string $i_{1}i_{2}\ldots i_{\ell-1}$, then $w_{\bf i}=e_{i_{\ell}0}^{p^{(\ell)}-1}w_{{\bf i}^{-}}$ and so, for all $n\in\mathbb{Z}\setminus\{0\}$, $0=h_{i_{\ell}n}e_{i_{\ell}0}^{p^{(\ell)}}w_{{\bf i}^{-}}=2p^{(\ell)}e_{i_{\ell}n}w_{\bf i}$, i.e., $e_{i_{\ell}n}w_{\bf i}=0$. Now, $0=h_{j0}e_{jm}e_{i_{\ell}0}^{p^{(\ell)}}w_{{\bf i}^{-}}=e_{jm}h_{j0}e_{i_{\ell}0}^{p^{(\ell)}}w_{{\bf i}^{-}}+2e_{jm}e_{i_{\ell}0}^{p^{(\ell)}}w_{{\bf i}^{-}}=2p^{(\ell)}e_{jm}e_{i_{\ell}0}^{p^{(\ell)}-1}w_{{\bf i}^{-}}=2p^{(\ell)}e_{jm}w_{\bf i}$.

Pick one of the non-zero $w_{\bf i}$ constructed above and let $W_{\bf i}=U(G)w_{\bf i}$ be a $G$-submodule of $M$. By construction $e_{ln}W_{\bf i}=0$ for all $l\in I_{0}$ and $n\in\mathbb{Z}$. Considered the induced module $I(W_{\bf i})=\operatorname{Ind}_{G\oplus H\oplus N_{+}}^{\hat{\mathfrak{g}}}W_{\bf i}$, where $N_{+}=\bigoplus_{i\in I_{0},n\in Z}\mathbb{C}e_{in}$ acts by $0$, $H=\bigoplus_{i\in I_{0}}\mathbb{C}h_{i}\oplus\mathbb{C}d$ acts by $h_{i}w_{\bf i}=\mu(h_{i})w_{\bf i}$, $dw_{\bf i}=\mu(d)w_{\bf i}$, for some weight $\mu$. Because $M$ is simple, it is a quotient of $I(W_{\bf i})$. If $w_{\bf i}\in T(M)$, we have $W_{\bf i}=\mathbb{C}w_{\bf i}$, and so $M$ is a quotient of $I(W_{\bf i})=\tilde{M}(\lambda)$ and we are done.

In case $w_{\bf i}\notin T(M)$, as in the proof of Proposition 6.0.3. of [2] we get a contradiction. This completes the proof.

Proof. Because $M$ is in $\mathcal{O}_{red,im}$, it is a $G$-compatible and so, it has a decomposition as a $G$-module given by $M\cong T(M)\oplus TF(M)$. Since all the weights of $M$ are in $\hat{\mathfrak{h}}^{*}_{red}$, $T(M)$ is not a $\hat{\mathfrak{g}}$-submodule of $M$. Indeed, suppose $T(M)$ is a $\hat{\mathfrak{g}}$-module. let $v\in T(M)$ and consider $f_{0}v\in T(M)$. Then $h_{0m}f_{0}v=0$ and $f_{m}v=0$ for any $m\neq 0$. Applying $h_{0,-m}$ we get $h_{0,-m}f_{m}v=0$ and $f_{0}v=0$. Since the weight of $v$ is in $\hat{\mathfrak{h}}^{*}_{red}$, $e_{0}^{p}v\neq 0$ for any $p>0$. But if $p$ is sufficiently large the weigh of $e_{0}^{p}v$ will not be in $\hat{\mathfrak{h}}^{*}_{red}$ and we get a contradiction.

Let $v\in T(M)$ non-zero. As in the proof of the previous statement there exists a string ${\bf i}$ of elements of $I_{0}$ and a vector $w_{\bf i}$ such that $e_{jm}w_{\bf i}=0$ for all $j\in I_{0}$ and $m\in\mathbb{Z}$. Let $W_{\bf i}=U(G)w_{\bf i}$. Then we have two possibilities: either $w_{\bf i}\notin T(M)$ or $w_{\bf i}\in T(M)$.

In the first case, consider the induced module $I(W_{\bf i})$. Clearly $TF(I(W_{\bf i}))\subseteq I(W_{\bf i})$. Now, if $w\in I(W_{\bf i})$, because $w_{\bf i}\notin T(M)$ we have $gw\neq 0$ for $g\in G$ and so $w\in TF(I(W_{\bf i}))$. Then $TF(I(W_{\bf i}))=I(W_{\bf i})$. By the five lemma, any quotient and subquotient of $I(W_{\bf i})$ also satisfies this property. Set $M^{\prime}:=U(\hat{\mathfrak{g}})w_{\bf i}$ which is a subquotient of $I(W_{\bf i})$. Then $M^{\prime}$ is a $\hat{\mathfrak{g}}$-submodule of $M$ and so $M^{\prime}=TF(M^{\prime})$ is a $\hat{\mathfrak{g}}$-submodule of $TF(M)$, but $TF(M)$ does not have proper $\hat{\mathfrak{g}}$-submodule and so $M^{\prime}=TF(M)$. But, $W_{\bf i}$ is a proper $G$-submodule of $M^{\prime}$ which is not possible because $M$ is in $\mathcal{O}_{red,im}$. And so, this case does not occur.